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Homework #9 Solutions - Trinity University
Homework #9 Solutions p 149, #18 Let n > 1. Then n−1 ∈ U(n) and (n−1)2 = n2−2n+1 so that (n−1)2 mod n = 1. Since n − 1 6= 1, this means that |n − 1| = 2 in U(n). As the order of any element in a group must divide the order of that group, it follows that 2 must divide the order of U(n), i.e. the order of U(n) is even.
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